Continuity and Level Sets of the Minkowski Gauge

If 0 lies in the interior of a convex set, its gauge is continuous and recovers int(Ω) and cl(Ω).

Continuity and Level Sets of the Minkowski Gauge

Let $X$ be a normed vector space and let $\Omega\subset X$ be convex with $0\in\operatorname{int}(\Omega)$ (see interior ).

Corollary: The Minkowski gauge $p_\Omega$ is continuous. Moreover,

\operatorname{int}(\Omega)=\{x\in X\mid p_\Omega(x)<1\}

and

\overline{\Omega}=\{x\in X\mid p_\Omega(x)\le 1\}.

Here $\overline{\Omega}$ denotes the closure of $\Omega$ .

Context: Continuity comes from the inclusion of a norm ball into $\Omega$ (since $0\in\operatorname{int}(\Omega)$ ), yielding a Lipschitz bound for $p_\Omega$ . The set identities combine the gauge level-set theorem with lin(Ω)=cl(Ω) for solid convex sets .